Biot-JKD model: Simulation of 1D transient poroelastic waves with fractional derivatives

  • Authors:
  • Emilie Blanc;Guillaume Chiavassa;Bruno Lombard

  • Affiliations:
  • Laboratoire de Mécanique et d'Acoustique, UPR 7051 - CNRS, 31 chemin Joseph Aiguier, 13402 Marseille, France;Centrale Marseille and M2P2, UMR 7340 - CNRS, Technopôle de Chateau-Gombert, 38 rue Frédéric Joliot-Curie, 13451 Marseille, France;Laboratoire de Mécanique et d'Acoustique, UPR 7051 - CNRS, 31 chemin Joseph Aiguier, 13402 Marseille, France

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2013

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Abstract

A time-domain numerical modeling of Biot poroelastic waves is presented. The viscous dissipation occurring in the pores is described using the dynamic permeability model developed by Johnson-Koplik-Dashen (JKD). Some of the coefficients in the Biot-JKD model are proportional to the square root of the frequency: in the time-domain, these coefficients introduce order 1/2 shifted fractional derivatives involving a convolution product. Based on a diffusive representation, the convolution kernel is replaced by a finite number of memory variables that satisfy local-in-time ordinary differential equations. Thanks to the dispersion relation, the coefficients in the diffusive representation are obtained by performing an optimization procedure in the frequency range of interest. A splitting strategy is then applied numerically: the propagative part of Biot-JKD equations is discretized using a fourth-order ADER scheme on a Cartesian grid, whereas the diffusive part is solved exactly. Comparisons with analytical solutions show the efficiency and the accuracy of this approach.